(3) One of the main difficulties in studying the existence of such weak solutions is the weak \(L^1\)  convergence of the pressure. Since the structure regularity that comes from the energy is not strong enough to ensure that the fluid cavity is Lipschitz (and it also changes in time), the standard approach based on Bogovskii operator fails in this framework. The alternative here consists in proving the additional integrability of the pressure inside the domain in the standard way and that the \(L^1\)  norm of the pressure doesn't concentrate near the boundary, thus giving the equiintegrability sufficient to pass to the weak limit. The latter proof is very involved and it is based on constructing a test function which sort of represents the distance with respect to the boundary. As a consequence, this construction in [1]} and [2]} varies greatly due to different geometries, as the latter has corners at \(\partial \Gamma \times \lbrace -1\rbrace \)  and \( \partial \Gamma \times \lbrace 0\rbrace \) . If \(\alpha =0\) , in order to close the estimates in the mentioned proof, one needs the condition \(\gamma >12/7\)  to compensate for the low integrability of the structure velocity \(\partial _t w \in L_t^2 L_x^p\) , for any \(p<4\) , which comes from the trace regularity. If however \(\alpha >0\) , then one can use the regularity \(\partial _t w \in L_t^2 H_x^1\) , so \(\gamma >3/2\)  suffices. This means that the same results as in [1]}, [2]} hold for \(\alpha >0\)  and \(\gamma >3/2\) , even though they weren't explicitly considered.

